Radius of comparison and mean cohomological independence dimension
aa r X i v : . [ m a t h . OA ] S e p RADIUS OF COMPARISON AND MEANCOHOMOLOGICAL INDEPENDENCE DIMENSION
ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
Abstract.
We introduce a notion of mean cohomological independencedimension for actions of discrete amenable groups on compact metriz-able spaces, as a variant of mean dimension, and use it to obtain lowerbounds for the radius of comparison of the associated crossed product C ˚ -algebras. Our general theory, gives the following for the minimalsubshifts constructed by Dou in 2017. For any countable amenablegroup G and any polyhedron Z , Dou’s subshift T of Z G with densityparameter ρ satisfiesrc p C p X q ⋊ T G q ą
12 mdim p T q ˆ ´ ´ ρρ ˙ ´ . If k “ dim p Z q is even and q H k p Z ; Q q ‰
0, thenrc p C p X q ⋊ T G q ą
12 mdim p T q ´ , regardless of what ρ is. The notion of mean dimension was outlined by Gromov in [Gro99], andlater fleshed out in a paper of Lindenstrauss and Weiss [LW00]. The generalphilosophy outlined in Gromov’s paper was that given an invariant Inv p X q for spaces X , one can try to define a dynamical variant Inv p X ; G q for actionsof groups on X , which should, as a test case, for the full shift roughly satisfyInv p X G ; G q “ Inv p X q . Of course, various restrictions may be placed on thespaces, on the groups, or on the actions. Entropy, for instance, can bethought of as a dynamical way to count cardinality. The mean dimensionmdim p X, G q is a dynamical variant of covering dimension. For actions of Z ,see Definition 2.6 of [LW00]; for amenable groups, see the remarks afterthis definition and the discussion of this case in the introduction to [LW00].One of the motivating applications was to show that not every dynamicalsystem of the form p X, Z q can be embedded into the full shift on r , s Z . If T denotes the action of G on X , we sometimes write mdim p T q in place ofmdim p X, G q .The notion of radius of comparison for C ˚ -algebras was introduced byToms in [Tom06], as a way to systematize the counterexamples to the El-liott program he constructed in [Tom08], based on techniques introducedfirst by Villadsen in [Vil98]. Let A be a unital stably finite C ˚ -algebra. Let τ be a tracial state on A . By slight abuse of notation, we also use τ todenote the induced trace on M p A q . For a positive element a P M n p A q , weset d τ p a q “ lim n Ñ8 τ p a { n q . For r ą
0, we say that A has r -comparison Date : 27 September 2020.This research was supported by Israel Science Foundation grant 476/16 and the SimonsFoundation Collaboration Grant for Mathematicians if for any two positive elements a, b P M p A q , if d τ p a q ` r ă d τ p b q for alltracial states τ on A then a - b ( a is Cuntz-subequivalent to b ). (In general,one should use quasitraces here, but the C ˚ -algebras in this paper will benuclear, so that all quasitraces are tracial states by [Haa14].) The radius ofcomparison of A is the infimum of all r ą A has r -comparison.Toms’ counterexample is of a simple AH algebra which has positive radiusof comparison but otherwise has the same Elliott invariant as an AI algebra(which has zero radius of comparison). Recent major advances in the studyof classification theory for nuclear C ˚ -algebras ([EGLN15, TWW17]), build-ing on decades of work by many authors, show that simple nuclear unital C ˚ -algebras satisfying the Universal Coefficient Theorem are classified viathe Elliott invariant provided they have finite nuclear dimension. Conjec-turally, this corresponds to the case of zero radius of comparison; this hasbeen proved when the tracial state space of A is a Bauer simplex whoseextreme boundary has finite covering dimension ([KR14, Sat12, TWW15]).The connection to dynamical systems was broached by Giol and Kerrin [GK10], where they constructed examples of minimal homeomorphismswhose crossed products have positive radius of comparison. The examples inthe paper of Giol and Kerr have positive mean dimension. That the spacesthemselves had to be infinite dimensional follows from the fact that for min-imal homeomorphisms of finite dimensional spaces, the crossed product hasfinite nuclear dimension and hence has zero radius of comparison ([TW13];see also [HWZ15]). This suggested a connection between mean dimensionand radius of comparison, two notions which came about independently andin different contexts. It has been conjectured by the second named authorand Toms that for minimal systems, the radius of comparison should be halfthe mean dimension. The second named author showed in [Phi16] that forminimal actions T of the integers, the radius of comparison of the crossedproduct is bounded above by 1 ` p T q . Elliott and Niu showed in[EN17] that for minimal actions of the integers, mean dimension zero im-plies zero radius of comparison. Recently, Niu ([Niu19b, Niu19a]) improvedthose results and showed that for free and minimal actions of Z d , the radiusof comparison of the crossed product is at most half the mean dimension.Those results mark very significant progress on this problem, but they allinvolve bounding the radius of comparison from above. For lower bounds,the only results we are aware of to date are for the examples constructed inthe paper of Giol and Kerr.The goal of this paper is to establish lower bounds (Theorem 3.3 and The-orem 4.5). In the case of commutative C ˚ -algebras, lower bounds for theradius of comparison of C p X q were obtained in [EN13] in terms of rationalcohomological dimension rather than covering dimension. For t P R , we de-note by t t u the greatest integer n such that n ď t . When covering dimensionand rational cohomological dimension coincide, the radius of comparison of C p X q is t dim p X q{ u ´ t dim p X q{ u ´
2; it is not known whether the lattercan occur. We refer to [Dra05] for a survey of cohomological dimension.The work of Elliott and Niu [EN13] suggests that, in order to obtain lowerbounds in the dynamical context, rather than using the Lindenstrauss-Weissnotion of mean dimension, which is based on covering dimension, we might
EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 3 look for a notion of “mean cohomological dimension”. Recall that a compactmetrizable space X has rational cohomological dimension d if d is the leastinteger such that for any k ą d and for any closed subset Y Ă X , we have q H k p X, Y ; Q q “ X , for any k we canask whether there exists a closed subset Y of X with non-vanishing k -thrational cohomology. We could define a notion of the dimension of X asthe supremum of all k P N such that this holds. Such a notion does notquite coincide with covering dimension for CW complexes. (For instance,the dimension of the three dimensional ball would be 2 rather than 3.)However, in the context of mean dimension, it sometimes does not matter ifthe dimension it is based on is off by a constant. For technical reasons, weactually consider only even integers k . This is related to the fact that wework with complex vector bundles and Chern classes; more philosophically,it reflects the fact that the radius of comparison should be thought of as asort of complex dimension rather than real dimension, which explains thefactor of 1 { k , we could ask whether there exists a subspace withnon-vanishing k -th rational cohomology. We think of cohomology classes η , η , . . . , η n as being “independent” if their cup product is nonzero. Roughlyspeaking, given an action of an amenable group G on X , and given a co-homology class η of a subspace, for any finite set F of G , we can find thelargest subset F such that the iterates of η under F are independent inthis sense, and then measure the upper density of such sets in a Følnersequence. This is used as a basis for our notion of mean cohomological inde-pendence dimension (Definition 1.11). For full shifts on a CW complex Z ,under a mild condition on the group, our dynamical invariant recovers thedimension of Z , thereby meeting the rule of thumb suggested in Gromov’spaper [Gro99]. There is a related but somewhat different notion of meanhomological dimension in Section 2.6.3 of [Gro99]. It applies specifically tosubshifts, and is used there to get lower bounds on the mean dimension.The reader may also find some analogy between the connection of meandimension with our notion of independence dimension and the connectionof entropy with combinatorial independence which was studied by Kerr andLi ([KL07]), although we do not use it in any way in this paper.In Section 1, we give some preliminaries and define the mean cohomolog-ical independence dimension. Section 2 contains estimates of the mean co-homological independence dimension of shifts and certain kinds of subshifts.Section 3 contains the main theorem, giving a lower bound on rc p C p X q ⋊ T G q in terms of mean cohomological independence dimension. When applied toa minimal subshift T of the shift on Z G , as constructed in [Dou17], using apolyhedron Z and density parameter ρ , this result implies (Corollary 3.4)rc p C p X q ⋊ T G q ą
12 mdim p T q ˆ ´ ´ ρρ ˙ ´ . Up to an additive constant, this estimate is close to the conjectured value mdim p T q when ρ is close to 1, but is useless if ρ ď . In Section 4, we ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS introduce a variant of our definition, which we call symmetric mean coho-mological independence dimension . This involves a stronger independencecondition, which allows us to obtain improved bounds for certain dynam-ical systems, such as subshifts of p S k q G for k even. In particular, for theconstruction of [Dou17] in this case, one gets (Corollary 4.6)rc p C p X q ⋊ T G q ą
12 mdim p T q ´ , regardless of the value of ρ , which is useful whenever ρ ą k . In Section 5,we state some open problems.1. Mean cohomological independence dimension
We begin by fixing some notation. Throughout this paper, X is a com-pact metrizable space, G is a countable amenable discrete group, and T isan action of G on X . When needed, we let α : G Ñ Aut p C p X qq be thecorresponding action of G on C p X q , given by α g p f qp x q “ f p T ´ g p x qq . Weusually write the crossed product C p X q ⋊ α G as C p X q ⋊ T G . (Since G isamenable, the full and reduced crossed products are the same.) Definition 1.1.
Let X be a compact metrizable space, and let Y Ă X beclosed. A finite open cover of Y in X is a finite collection of nonempty opensubsets of X whose union contains Y . We often omit mention of X when itis understood. We denote by N p U q the nerve of U .To emphasize: a finite open cover of Y consists of subsets of X which areopen in X , not of open subsets of Y . For this and the following definitions(but not for some of the lemmas), there is no reason not to use arbitrarytopological spaces X and arbitrary subsets Y .We exclude ∅ from covers to avoid later improperly claiming that U Yt ∅ u “ U .The less convenient alternative is to work with finite open covers of variousclosed sets Y in the traditional sense. The outcome will be the same; seeLemma 1.6 and Lemma 1.7 below.We now give definitions which are standard for open covers, slightly mod-ified for our present situation. Definition 1.2.
Let X be a compact metrizable space, and let U and U be collections of nonempty open sets in X . Then their join is U _ U “ U X U | U P U , U P U , and U X U ‰ ∅ ( . By iteration, we get the join of any finite set of collections of open sets. If U , U , . . . , U n are finite open covers in X of closed subsets Y , Y , . . . , Y n Ă X , then U _ U _ ¨ ¨ ¨ _ U n is a finite open cover of Y X Y X ¨ ¨ ¨ X Y n in X . Definition 1.3.
Let X be a compact metrizable space, let Y Ă X be closed,and let U and V be finite open covers of Y . Then V refines U (as a coverof Y ; written V ă Y U ) if for every V P V there is U P U such that V Ă U .Formally, the only role that Y plays is that we are restricting to finiteopen covers of Y . EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 5
Definition 1.4.
Let X be a compact metrizable space, let Y Ă X be closed,and let U be a finite open cover of Y in X . The order of U isord p U q “ max x P X Card ` t U P U | x P U u ˘ ´ . We denote by D Y p U q the least order of any refinement of U among finiteopen covers of Y in X .In the situation of Definition 1.4, the order of U is the dimension of N p U q .While the subset Y is formally irrelevant in Definition 1.3, the quantity D Y p U q depends strongly on Y . Notation 1.5.
Let X be a compact metrizable space, let Y Ă X be closed,and let U be a finite open cover of Y in X . We set U X Y “ U X Y | U P U and U X Y ‰ ∅ ( , which is a finite open cover of Y , regarded as a topological space in its ownright. Lemma 1.6.
Let X be a compact metrizable space, let Y Ă X be closed,and let V be a finite open cover of Y , regarded as a topological space inits own right. Then there is a finite open cover W of Y in X such that ord p W q ď ord p V q and W X Y refines V in the conventional sense for opencovers of Y in Y .Proof. Write V “ t V , V , . . . , V n u with V , V , . . . , V n distinct. Choose opensubsets U , U , . . . , U n Ă X such that V j “ U j X Y for j “ , , . . . , n . Chooseopen subsets U j,l Ă X for l “ , , . . . with U j, Ă U j, Ă U j, Ă U j, Ă ¨ ¨ ¨ Ă U j and ď l “ U j,l “ U j . Then Y Ă ď l “ ˜ n ď j “ U j,l ¸ , so, by compactness, there is l P t , , . . . u such that Y Ă Ť nj “ U j,l .Choose open subsets Z , Z , . . . Ă X such that Z Ą Z Ą Z Ą Z Ą ¨ ¨ ¨ Ą Y and č m “ Z m “ Y. Let S be the set of all subsets J Ă t , , . . . , n u such that Ş j P J V j “ ∅ . For J P S , we have č m “ ˜č j P J ` U j,l X Z m ˘¸ “ ∅ . Therefore there exists m J P t , , . . . u such that č j P J ` U j,l X Z m J ˘ “ ∅ . Define m “ max J P S m J . For j “ , , . . . , n define W j “ U j,l X Z m , and set W “ t W , W , . . . , W n u . One easily checks that W satisfies the conclusionof the lemma. (cid:3) ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
Lemma 1.7.
Let X be a compact metrizable space, let Y Ă X be closed,and let U be a finite open cover of Y in X . Then D Y p U q “ D Y p U X Y q . The expression U X Y is as in Notation 1.5, and D Y p U X Y q is what isusually called D p U X Y q , taken among finite open covers of Y as a topologicalspace in its own right. Proof of Lemma 1.7.
We first claim that D Y p U q ď D Y p U X Y q . Choose afinite open cover V of Y in Y such that V ă Y U X Y and ord p V q “ D Y p U X Y q .Use Lemma 1.6 to choose a finite open cover W of Y in X such that W X Y ă Y V and ord p W q ď ord p V q . For each W P W there is U W P U such that W X Y Ă U W X Y . Set W “ W X U W | W P W ( . Clearly W covers Y in X , W - Y U , and ord p W q ď ord p W q . So D Y p U q ď ord p W q ď ord p W q ď ord p V q “ D Y p U X Y q , proving the claim.For the reverse inequality, choose a finite open cover W of Y in X suchthat W ă Y U and ord p W q “ D Y p U q . Then D Y p U X Y q ď ord p W X Y q ď ord p W q “ D Y p U q , as desired. (cid:3) In particular, the covering dimension dim p Y q can be calculated using opencovers of Y in X instead of conventional open covers of Y . Definition 1.8.
Let X be a compact metrizable space, let Y Ă X beclosed, and let U be a finite open cover of Y in X . Let R be a commutativeunital ring. For k “ , , , . . . , we denote by q H k p Y ; U ; R q the set of thoseelements of the ˇCech cohomology group q H k p Y ; R q which can be representedby cocycles arising from U X Y (as in Notation 1.5).Suppose Y and Y are two closed subsets of X . Suppose η P q H k p Y ; R q and η P q H m p Y ; R q . Though we cannot define the cup product of these twoelements, as they belong to different groups, we can restrict them to theintersection and consider the cup product η | Y X Y ! η | Y X Y P q H k ` m p Y X Y ; R q . Lemma 1.9.
Let X be a compact metrizable space, let Y , Y Ă X be closed,let R be a commutative unital ring, and for j “ , let U j be a finite opencover of Y j in X , let m j P t , , , . . . u , and let η j P q H m j p Y j ; U j ; R q . Then η | Y X Y ! η | Y X Y P q H m ` m p Y X Y ; U _ U ; R q . Proof.
Set Y “ Y X Y , and set V j “ U j X Y , a finite open cover of Y . Itsuffices to prove that if η j P q H m j p Y ; V j ; R q for j “ ,
2, then η ! η P q H m ` m p Y ; V _ V ; R q . Since V _ V refines both V and V , we have η j P q H m j p Y ; V _ V ; R q for j “ ,
2. This implies that there exist a map h : Y Ñ N p V _ V q and elements ξ j P q H m j p N p V _ V q ; R q such that η j “ h ˚ p ξ j q for j “ ,
2. Thus, ξ ! ξ P q H m ` m p N p V _ V q ; R q , andbecause η ! η “ h ˚ p ξ ! ξ q , we have η ! η P q H m ` m p Y ; V _ V ; R q ,as claimed. (cid:3) EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 7 If F is a (finite) set, we denote by Card p F q the cardinality of F . Definition 1.10. If G is a group, F Ă G is a nonempty finite subset, G Ă G , and δ ą
0, we say that F is p G , δ q -invariant if Card p gF X F q ąp ´ δ q Card p F q for all g P G .By convention, p G , δ q -invariant subsets are nonempty. Definition 1.11.
Let X be a compact metrizable space, let G be a countableamenable group, and let T be an action of G on X . Let R be a commutativeunital ring. For any even integer k , we define mcid k p T ; R q to be the largest d P r , such that the following happens.For every ε ą Y Ă X , a finite open cover U of Y in X such that D Y p U q P t k, k ` u , and η P q H k p Y ; U ; R q (Definition 1.8:ˇCech classes using U ) such that for every finite subset G Ă G and every δ ą p G , δ q -invariant nonempty finite set F Ă G and a subset F Ă F for which the following happen:(1) The cup product of T ˚ g p η q over all g P F , which makes sense as anelement of q H k ¨ Card p F q `Ş g P F T ´ g p Y q ; R ˘ , is nonzero.(2) k ¨ Card p F q Card p F q ą d ´ ε .We then say that T has mean k -th cohomological independence dimension d with coefficients in R .We define the mean cohomological independence dimension mcid p T ; R q tobe the supremum of mcid k p T ; R q over all k P t , , , , . . . u . Remark 1.12.
The empty cup product is taken to be 1. It is then easy tocheck that the set of d singled out by the second paragraph of the definition isan interval of the form r , r s for some r P r , k s . In particular, mcid k p T ; R q ď k . Remark 1.13.
We defined mean cohomological independence dimensionwith coefficients in an arbitrary commutative unital ring R , but in this paperwe only use the case R “ Q . One could think of various generalizations inwhich one replaces ˇCech cohomology by more general sheaf cohomology.However we do not have any use for that here.We took k to be even for technical reasons which will become apparentlater. Essentially, this is as we intend to work with complex vector bundles,and in some sense our notion of dimension should be thought of as complexdimension. We could have made the definition without this restriction,noting that, as the cup product would no longer be commutative, we wouldhave to make an arbitrary choice for the order in which the product istaken. We do not know whether a version which allows odd values of k issignificantly different. Remark 1.14.
It is immediate from the Definition 1.11 that if T is anaction of G on a compact metrizable space X and Y is a closed invariantsubset, then mcid k p T | Y ; R q ď mcid k p T ; R q for any k , and mcid p T | Y ; R q ď mcid p T ; R q . ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS Mean cohomological independence dimension of subshifts
In this section, we give estimates on mcid p T ; R q when T is a shift, or asubshift of the type considered in [GK10], [Kri09], and [Dou17]. We startwith a general result: mcid k p T ; R q ď mdim p T q . Proposition 2.1.
Let X be a compact metrizable space, let G be a countableamenable group, let T be an action of G on X , and let R be a commutativeunital ring. Then for any even natural integer k , we have mcid k p T ; R q ď mdim p T q .Proof. Without loss of generality mdim p T q ă 8 .Fix an even integer k , and let ε ą
0. Pick a compact subset Y of X , afinite open cover U of Y in X such that D Y p U q P t k, k ` u , and an element η P q H k p Y ; U ; R q such that for any finite subset G Ă G and any δ ą p G , δ q -invariant subset F Ă G and a subset F Ă F for which:(1) ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q ‰ k ¨ Card p F q Card p F q ą mcid k p T ; R q ´ ε .Recall that U may not be a cover of X . To remedy that, set U “ U Yt X r Y u . By the definition of mdim p T q , there are a finite subset G Ă G and δ ą p G , δ q -invariant subset F Ă G we have D X `Ž g P F T g ´ p U q ˘ Card p F q ă mdim p T q ` ε. Choose sets F and F as above to go with this choice of G . Set Z “ č g P F T g ´ p Y q and Z “ č g P F T g ´ p Y q . Further define W “ ł g P F T g ´ p U q , W “ ł g P F T g ´ p U q , and W “ ł g P F T g ´ p U q . (We don’t need the cover that would logically be called W .) Then W and W are open covers of X and W is an open cover of Y in X .We claim that, following Notation 1.5, we have W X Z “ W X Z . Toprove the claim, first recall that W X Z is the set of nonempty sets in g P F ` T ´ g p U g q X Z ˘ | U g P U for g P F + . In W X Z , we must also allow U g “ X r Y for some values of g P F , butthe additional sets gotten this way are all empty. The claim is proved.We have (justifications afterwards) D X p W q ě D Z p W q ě D Z p W q “ D Z p W q . The first inequality follows from Z Ă X , the second from W ă Z W , andthe third from the claim above and two applications of Lemma 1.7.Now, ! g P F T ˚ g p η q| Z P q H k Card p F q p Z ; W ; R q , EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 9 and therefore, in particular, q H k Card p F q p Z ; W ; R q ‰
0. As the elementsin this cohomology group can be realized as pullbacks of elements fromnerves of arbitrary refinements of W X Y , it follows from Lemma 1.7 that D Z p W q ě k Card p F q . Thereforemdim p T q ` ε ą D X p W q Card p F q ě k Card p F q Card p F q ě mcid k p T ; R q ´ ε. As ε ą p T q ě mcid k p T ; R q , as required. (cid:3) Recall that if G is a discrete group and Z is a set, then the shift actionof G on Z G is given by T g p x q h “ x g ´ h for any x “ p x g q g P G P Z G and all g, h P G .For the remainder of this section, we add the assumption that R is aprincipal ideal domain. This is done because the properties of cup productswhich we need are often derived in the context of the K¨unneth Formula.While this requirement is not strictly needed for the estimates in this section,we have no present use for such a possible generalization. Proposition 2.2.
Let Z be a finite CW complex, and let G be a countableamenable group. Suppose X “ Z G and let T be the shift. Let R be a principalideal domain.(1) For any even k ă dim p Z q we have mcid k p T ; R q “ k .(2) We have mcid p T ; R q ě Z dim p Z q ´ ^ . (3) If furthermore G has a quotient which is infinite and residually finite,then mcid p T ; R q “ dim p Z q . The hypothesis in (3) is equivalent to saying that G has subgroups ofarbitrarily large finite index. Proof of Proposition 2.2.
We prove (1). Let k be an even integer such that k ă dim p Z q . As Z has a cell of dimension greater than k , we can embed thesphere S k into Z . To simplify notation, fix a copy of S k in Z , and simplywrite S k Ă Z . Let q : Z G Ñ Z be the projection onto the coordinate g “ Y “ q ´ p S k q . Fix an isomorphism q H k p S k ; R q Ñ R . Let η P q H k p S k ; R q be the element mapped to the identity of R under this isomorphism. Choosea finite open cover V of S k for which q H k p S k ; V ; R q Ñ q H k p S k ; R q is anisomorphism. Choose a finite open cover V of S k in Z such that V X S k “ V .Then q H k p S k ; V ; R q “ q H k p S k ; V ; R q , so η P q H k p S k ; V ; R q . Set U “ q ´ p V q ,which is a finite open cover of Y in Z G , and set η “ p q | Y q ˚ p η q P q H k p Y ; V ; R q .We have D S k p V q “ k , so D S k p V q “ k by Lemma 1.7. It is now easily seenthat D Y p U q ď k , and the reverse inequality follows from q H k p Y ; V ; R q ‰ F Ă G is finite, then ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q ‰ . This will imply that mcid k p T ; R q ě k . Since mcid k p T ; R q ď k by Re-mark 1.12, it follows that mcid k p T ; R q “ k . To prove the claim, choose any point y P Z G r F . Identify č g P F T g ´ p Y q “ p S k q F ˆ Z G r F . Define maps t : p S k q F Ñ č g P F T g ´ p Y q and p : č g P F T g ´ p Y q Ñ p S k q F by t p x q “ p x, y q for x P p S k q F and p p x, y q “ x for x P p S k q F and y P Z G r F .Then p ˝ t “ id p S k q F . Set µ “ ź g P F η P q H k Card p F q ` p S k q F ; R ˘ . (The order in the product does not matter because k is even.) Then µ is agenerator of this group and in particular is nonzero. Naturality implies that ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q “ p ˚ p µ q . Now t ˚ p p ˚ p µ qq “ µ ‰
0, so p ˚ p µ q ‰
0. This is the claim, and part (1) isproved.Part (2) is immediate from part (1).We prove (3). The case dim p Z q “ p Z q ą
0. Since G has a quotient which is infinite but residually finite, G has arbitrarily largefinite quotients. For a finite set S Ă G , we denote by q S : Z G Ñ Z S theprojection onto the coordinates given by S . Note that Z S is a CW complexof dimension Card p S q ¨ dim p Z q .Fix δ ą
0. We will prove that there exists an even integer k such thatmcid k p T ; R q ą dim p Z q ´ δ .Pick a normal group N ⊳ G of finite index such that r G : N s ą { δ . Let S Ă G be a set of coset representatives for G { N , with 1 P S . Let k be thelargest even integer less than Card p S q ¨ dim p Z q “ dim p Z S q . As before, butwith Z S in place of S , fix an embedding of S k into Z S and a finite opencover V of S k in Z S such that q H k p S k ; V , R q “ q H k p S k ; R q , set Y “ q ´ S p S k q ,fix an isomorphism q H k p S k ; R q Ñ R , and let η P q H k p S k ; R q correspond to1 P R . Then η P q H k p S k ; V ; R q . Set U “ q ´ S p V q , and set η “ p q S | Y q ˚ p η q .As before, we have D Y p U q “ k .Let G Ă G be a finite set and let ε ą
0. Define M “ N X s ´ gt | g P G and s, t P S ( , which is a finite subset of N . Since N is amenable, there is a finite nonempty p M, ε q -invariant subset F Ă N . Define F “ SF . Then Card p F q “ Card p S q Card p F q . We claim that F is p G , ε q -invariant. So let g P G .For each t P S there is a unique s p g, t q P S such that gt P s p g, t q N . Then gtF Ă s p g, t q N , so gtF X F Ă s p g, t q N , whence gtF X F “ gtF X s p g, t q F .Now s p g, t q ´ gt P M , soCard p gtF X F q “ Card p gtF X s p g, t q F q“ Card p s p g, t q ´ gtF X F q ą p ´ ε q Card p F q . EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 11
One can check that if t , t P S and s p g, t q “ s p g, t q , then t ´ t P N ,whence t “ t . Therefore the sets gtF X F , for t P S , are disjoint. Summingover all t P S now givesCard p gtF X F q “ ÿ t P S Card p gtF X s p g, t q F qą p ´ ε q Card p S q Card p F q “ p ´ ε q Card p F q , as claimed.Since the translates of S under F are disjoint, reasoning similar to thatused in the proof of part (1) shows that ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q ‰ . Also k ¨ Card p F q Card p F q ě “ Card p S q dim p Z q ´ ‰ ¨ Card p F q Card p F q“ dim p Z q ´ p S q ą dim p Z q ´ δ. It follows that mcid k p T ; R q ě dim p Z q ´ δ , as desired.Since δ ą p T ; R q ě dim p Z q . On the other hand,it is easy to deduce from Corollary 4.2 of [CK05] that mdim p T q ď dim p Z q . Itnow follows from Proposition 2.1 that mcid p T ; R q “ dim p Z q , as required. (cid:3) One of the main sources of examples of minimal homeomorphisms withnonzero mean dimension is subshifts. Krieger established some lower boundsin [Kri09], generalizing the case of Z which was discussed in [LW00], and Douin [Dou17] constructed more specific examples in which one can computemean dimension precisely. Here we obtain a related lower bound for meancohomological independence dimension for subshifts.For convenience, we give two definitions. The first is standard. Thesecond is intended only for use in this paper, and identifies a feature whichis a useful hypothesis and which is common among constructions in theliterature of minimal subshifts. Definition 2.3.
Let G be a countable amenable group. A Følner sequence in G is a sequence p F n q n P N of nonempty finite subsets F n Ă G such that forall g P G , we have lim n Ñ8 Card p gF n △ F n q Card p F n q “ . For a subset J Ă G , we define its density δ p J q to be the supremum over allFølner sequences p F n q n P N in G oflim sup n Ñ8 Card p J X F n q Card p F n q . Definition 2.4.
Let G be a discrete group, let Z be a set, and let T bethe shift action of G on Z G . Let X Ă Z G be T -invariant. We say that asubset J Ă G is X -unconstrained (or just unconstrained if X is understood)if there is a point z “ p z g q g P G P X such that for any x “ p x g q g P G P Z G , if x g “ z g for any g R J then x P X . We call such a point z a witness for the X -unconstrainedness of J . If X “ Z G then G itself is X -unconstrained. Subsets of unconstrainedsets in G are unconstrained. We are interested in shift invariant subsets X Ă Z G . In this case, if J is X -unconstrained, with witness z , and g P G ,then gJ is also X -unconstrained, with witness T g p z q . Proposition 2.5.
Let Z be a polyhedron, let ρ P p , q , and let X Ă Z G bethe minimal subshift of the shift T on Z G constructed in Section 4 of [Dou17] to satisfy mdim p T q “ ρ dim p P q . Then there is an X -unconstrained subset J Ă G such that δ p J q ě ρ .Proof. We use the set J Ă G constructed in Section 4.2 of [Dou17]. Itis proved there that δ p J q ě ρ . The second half of Section 4.2 of [Dou17]proves the existence of z “ p z g q g P G P X such that for any x “ p x g q g P G P Z G ,if x g “ z g for any g R J then x P X , that is, z is a witness for the X -unconstrainedness of J . (cid:3) The fact that the set J in [Dou17] is X -unconstrained is crucial in theproof there of the lower bound for mdim p T q . The assumption in Proposition2.5 that Z is a polyhedron follows [Dou17]. However, we assume that thesame holds if we assume that Z is any finite CW-complex. Proposition 2.6.
Let Z be a finite CW-complex. Let X be a closed G -invariant subset of Z G , with the shift action T . Let J Ă G be an X -unconstrained subset with witness z “ p z g q g P G P X . Let ρ “ δ p J q be thedensity of J . Let R be a principal ideal domain. Then for any even integer k ă dim p Z q we have mcid k p T ; R q ě kρ . Moreover, mcid p T | X ; R q ě ρ Z dim p Z q ´ ^ . Proof.
It suffices to prove the first statement. The proof is similar to that ofProposition 2.2(1). We may assume dim p Z q ą
0. We may assume withoutloss of generality that 1 P J .Let k be a nonnegative even integer such that k ă dim p Z q . As Z hasa cell of dimension greater than k , we can embed the sphere S k into Z .As in the proof of Proposition 2.2(1), fix an embedding of S k into Z anda finite open cover V of S k in Z such that q H k p S k ; V , R q “ q H k p S k ; R q ,fix an isomorphism q H k p S k ; R q Ñ R , and let η P q H k p S k ; R q correspondto 1 P R . Then η P q H k p S k ; V ; R q . Let q : Z G Ñ Z be the projectiononto the coordinate g “
1. Then q p X q “ Z . Set Y “ q ´ p S k q X X , set U “ q ´ p V q X X , and set η “ p q | Y q ˚ p η q . As before, and relying on the nextclaim, we have D Y p U q “ k .We claim that for any finite set F Ă G , the element ν “ ! g P F X J T ˚ g p η q| Ş h P F X J T ´ h p Y q is nonzero. Letting F run through suitable Følner sequences, the definitionwill give mcid k p T | X ; r q ě kρ , as required.To prove the claim, define maps t : p S k q F X J Ñ č g P F X J T g ´ p Y q and p : č g P F X J T g ´ p Y q Ñ p S k q F X J EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 13 as follows. Take p to be the restriction of the projection map p S k q F X J ˆ Z G r p F X J q Ñ p S k q F X J . For x P p S k q F X J define t p x q g “ x g g P F X Jz g g P G r p F X J q . The conditions on z imply that t p x q as defined here really is in X , and itnow follows that t p x q P Ş g P F X J T g ´ p Y q . Then p ˝ t “ id p S k q F . Followingthe proof of Proposition 2.2(1), set µ “ ś g P F X J η , use naturality to get ν “ p ˚ p µ q , and use µ ‰ t ˚ p p ˚ p µ qq “ µ to get p ˚ p µ q ‰
0. This is theclaim. (cid:3)
Remark 2.7.
One can use Proposition 2.6, Remark 1.14, and Proposition2.2(3) to obtain obstructions for embedding various subshifts in full shifts.This does not entirely bypass the use of mean dimension, as we used meandimension in the proof of Proposition 2.2(3).3.
Lower bounds for the radius of comparison
The goal of this section is to show that for any countable amenable group G , the radius of comparison of C p X q ⋊ T G is bounded below by mcid k p T ; Q q´ ´ k { K p n, Q q theEilenberg-MacLane spaces, that is, the classifying spaces for ˇCech cohomol-ogy with coefficients in Q . This means that for a compact metrizable space Y , the ˇCech cohomology group q H n p Y ; Q q is naturally isomorphic to the set r Y, K p n, Q qs of homotopy classes of maps from Y to the Eilenberg-MacLanespace. The space K p n, Q q is unique up to homotopy equivalence. Remark 3.1.
The Chern classes of a vector bundle E over a compactmetrizable space X are elements c q p E q of q H q p X ; Z q . (See Theorem V.3.15and Remark V.3.21 of [Kar78].) The total Chern class is c p E q “ ` c p E q ` c p E q ` ¨ ¨ ¨ . When we work in rational cohomology, we use the rationalversion c Q q p E q “ c q p E q b Q P q H q p X ; Z q b Q – q H q p X ; Q q and, similarly, c Q p E q P q H ˚ p X ; Q q . Lemma 3.2.
Let Z be a finite CW complex. Suppose q is a positive integerand η P q H q p Z ; Q q is a nonzero cohomology class. Then there exists apositive integer m and a vector bundle E over Z such that the q -th rationalChern class of E is c Q q p E q “ mη and such that c Q j p E q “ for all j Pt , , . . . , q ´ u .Proof. By the construction of Eilenberg-MacLane spaces in Section 4.2 of[Hat02], we can choose a model for K p q ; Q q with no cell of dimensionstrictly between 0 and 2 q . Represent the cohomology class η as a function f : Z Ñ K p q ; Q q . Since Z is compact, the image of Z is contained in a finitesubcomplex Z of K p q ; Q q (by [Hat02, Proposition A.1]). Then Z also has no cells of dimension strictly between 0 and 2 q . Let ι : Z Ñ K p q ; Q q be theinclusion map, and let f : Z Ñ Z be the map f , restricting the codomainto be Z , so that f “ ι ˝ f . Denote by r ι s P q H q p Z ; Q q the class representedby ι . Then η “ p f q ˚ pr ι sq .We refer the reader to the proof of Proposition IV.7.11 of [Kar78] for thedefinition of the Newton polynomials Q n for n “ , , , . . . , and to SectionV.3 of [Kar78] for a discussion of the Chern character. If X is a compactmetrizable space and E is a vector bundle over X , recall (V.3.19 and V.3.22in [Kar78]) that we define the n -th component of the Chern character of avector bundle E byCh n p E q “ n ! Q n ` c Q p E q , c Q p E q , . . . , c Q n p E q ˘ P q H n p X ; Q q , and that Ch p E q “ ř n “ Ch n p E q (with Ch p E q taken to be the dimensionof E ). The Chern character gives rise to an isomorphism Ch : K p X q b Q Ñ q H even p X ; Q q (Theorem V.3.25 of [Kar78]).Returning to the situation in the first paragraph, we know that there existvector bundles F , F , . . . , F s over Z and rational numbers r , r , . . . , r s suchthat Ch q `ř sl “ r l r F l s ˘ “ r ι s . However, since Z has no cells of dimensionstrictly between 0 and 2 q , we know that c Q j pr F l sq “ j “ , , . . . , q ´ l “ , , . . . , s . Pick a strictly positive integer p such that pr l P Z for l “ , , . . . , s . We thus have Ch q p ř sl “ pr l r F l sq “ p r ι s . Some of thosecoefficients may be negative. To overcome that, for each l such that r l ă F l with its complement in a sufficiently large trivial bundle, andreplace r l with ´ r l . This change replaces ř sl “ pr l r F l s with r H s` ř sl “ pr l r F l s for some trivial bundle H . It has the effect of changing the value of Ch ,which we do not care about, but not the value of Ch q . We may thus assumewithout loss of generality that r l ą l “ , , . . . , s .The formal sum ř sl “ pr l r F l s can be replaced now by the direct sum, so wehave a vector bundle F over Z which satisfies Ch q p F q “ p r ι s and c Q j p F q “ j “ , , . . . , q ´
1. Thus, as all but one of the terms in the Newtonexpression vanish, we have (justification for the second step below)Ch q p F q “ q ! Q q ` c Q p F q , c Q p F q , . . . , c Q q p F q ˘ “ q ! ¨ p´ q q ´ qc Q q p F q “ p´ q q ´ p q ´ q ! c Q q p F q . The second step can be deduced from (2.11 ) on page 23 of [Mac95] byrearranging terms; see pages 19 and 23 there for the notation. Let F be thedirect sum of p q ´ q ! copies of F if q is odd, and the complement in a largetrivial bundle of the direct sum of p q ´ q ! copies of F if q is even. Since thetotal Chern class is multiplicative and c p F q “ ` c q p F q ` c q ` p F q ` ¨ ¨ ¨ ,we have c Q q p F q “ p q ´ q ! p r ι s . Now, set m “ p q ´ q ! p , and let E “p f q ˚ p F q . Then, by naturality, we have c Q q p E q “ mη and c Q j p E q “ j “ , , . . . , q ´ (cid:3) We can now prove our main theorem.
EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 15
Theorem 3.3.
Let G be a countable amenable group, let X be a compactmetrizable space, and let T be an action of G on X . Let k be an eveninteger, and let m be the greatest integer with m ă mcid k p T ; Q q . Then rc p C p X q ⋊ T G q ě m ´ k { . If C p X q ⋊ T G is simple, then rc p C p X q ⋊ T G q ą m ´ k { . In particular, rc p C p X q ⋊ T G q ě mcid k p T ; Q q ´ ´ k { Proof of Theorem 3.3.
Fix an even integer k . Fix ε ą Y Ă X , a finite open cover U of Y in X such that D Y p U q P t k, k ` u , and a cohomology class η P q H k p Y ; U ; Q q such that forany finite subset G Ă G and any δ ą p G , δ q -invariantsubset F and a subset F Ă F satisfying:(1) ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q ‰ k ¨ Card p F q Card p F q ą mcid k p T ; Q q ´ ε .Pick a finite open cover V of Y in X such that V ă Y U and ord p V q “ D Y p U q . Recall that N p V q is the nerve of V . By the construction of ˇCechcohomology, η is also in q H k p Y ; V ; Q q , and can be obtained as a pullback ofa cohomology class η P q H k p N p V q ; Q q by a map f : Y Ñ N p V q . Use Lemma3.2 to choose a vector bundle E over N p V q such that c Q k { p E q “ M η forsome nonzero integer M and c Q j p E q “ j P t , , . . . , k { ´ u . Sincedim p N p V qq ă k `
2, we also have c Q j p E q “ j P t k { ` , k { ` , . . . u .Therefore c Q p E q “ ` M η . As we can replace η by any nonzero scalarmultiple of it and retain the same properties, we may use M η in place of η , sowe may lighten notation and assume that c p E q “ ` η . Since dim p N p V qq ď k `
1, by subtracting trivial bundles (see Theorem 9.1.2 of [Hus94]), wecan also assume that rank p E q “ k {
2. Now set E “ f ˚ p E q . We have c p E q “ ` η by naturality, and rank p E q “ k { L be the dimension of some trivial bundle which has E as a direct sum-mand, and let q P M L p C p Y qq be the projection onto E . Let a P M L p C p X qq be a positive contraction such that a | Y “ q and rank p a p x qq ď rank p E q for all x P X . Let b P M p C p X qq be a constant projection. Supposethat a - C p X q ⋊ T G b . We are going to prove that this implies rank p b q ě mcid k p T ; Q q ´ ε .Increasing L if needed, we may assume that b P M L p C p X qq . There is c P M p C p X q ⋊ T G q such that } c ˚ bc ´ a } ă {
8. Replacing c by itscutdown by 1 M L p C p X q ⋊ T G q , we may assume that c P M L p C p X q ⋊ T G q – p M L b C p X qq ⋊ id ML b T G. Then there is c in the algebraic crossed product such that(3.1) } c ˚ bc ´ a } ă . With u g P C p X q ⋊ T G being the standard unitary corresponding to g P G ,there are a finite subset G Ă G and elements c g P M L p C p X qq for g P G such that c “ ř g P G c g p M L b u g q . Increasing the size of the set G , we may assume that(3.2) G “ g ´ | g P G ( . Choose δ ą δ ¨ p} c } ` q ¨ ÿ g P G } c g } ă . Choose a nonempty finite p G , δ q -invariant subset G of G with 1 P G .Fix ε ą k p T ; Q q ´ ε ´ kε ` ε ą mcid k p T ; Q q ´ ε . Set ε “ ε { Card p G q . Find a finite nonempty ` G Y p G q ´ , ε ˘ -invariantsubset F Ă G such that there exists a subset F Ă F for which:(3) ! g P F T ˚ g p η q| Ş h P F T ´ h p Y q ‰ k ¨ Card p F q Card p F q ą mcid k p T ; Q q ´ ε .For any h P G , if we replace F and F by F h ´ and F h ´ , then we replacethe class in (3) by its image under T ˚ h , which is still nonzero. Doing thiswith some h P F , we may therefore assume that 1 P F .Set K “ č h P G hF. Then Card p F r K q “ Card p F q ´ Card p K q ď ÿ h P G Card p F r r F X hF sqă Card p G q ε Card p F q “ ε Card p F q . (3.4)Let ∆ : G Ñ r , s be the function∆ p g q “ p G q p χ G ˚ χ F qp g q“ p G q ÿ s P G χ G p s q χ F p s ´ g q “ p G q ÿ h P G χ G p gh ´ q χ F p h q for g P G . Note that(3.5) ∆ p g q “ p G q ÿ h P F χ G p gh ´ q “ p G q ÿ s P G χ F p s ´ g q . Then for t P G and g P G we have, using (3.2) and p G , δ q -invariance of G at the last step, | ∆ p t ´ g q ´ ∆ p g q| “ ˇˇˇˇˇ p G q ÿ h P F ` χ g ´ tG p h ´ q ´ χ g ´ G p h ´ q ˘ˇˇˇˇˇ ď Card ` p tG r G q Y p G r tG q ˘ Card p G q ă δ. (3.6)We further claim that ∆ p g q “ g P K . To see this, notice that,by (3.5), we have ∆ p g q “ s ´ g P F for all s P G , that is, g P K . EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 17
Let α be the corresponding action of G on C p X q , that is, α g p f qp x q “ f p T ´ g p x qq for g P G , f P C p X q , and x P X . Let l p G q b C p X q be the usualHilbert C p X q -module, and write its elements as functions ξ : G Ñ C p X q such that ř g P G ξ p g q ˚ ξ p g q converges in C p X q . For any Hilbert module H ,let B p H q denote the C ˚ -algebra of adjointable operators on H . We view C p X q ⋊ T G as embedded in B p l p G q b C p X qq in the standard way, that is,if f P C p X q , ξ P l p G q b C p X q , and g, h P G , then p f ¨ ξ qp g q “ α g ´ p f q ¨ ξ p g q and p u h ¨ ξ qp g q “ ξ p h ´ g q . We define a multiplication operator d P B p l p G q b C p X qq by p d ξ qp g q “ ∆ p g q ξ p g q for g P G . By (3.6), for any t P G , } u t d u ˚ t ´ d } ă δ , whence } u t d ´ d u t } ă δ . Set d “ d b M L . It follows that(3.7) } cd ´ dc } ă δ ÿ g P G } c g } ă p} c } ` q . Notice that supp p ∆ q “ G F . Thus, we can view dM L p C p X q ⋊ T G q d asincluded in B ` l p G F q b M L b C p X q ˘ , with l p G F q b M L b C p X q regardedas a Hilbert M L b C p X q -module. Since G F is a finite set, this is a matrixalgebra over C p X q .Since, in particular, F is ` p G q ´ , ε ˘ -invariant, we have(3.8) Card p G F q ă ` ` ε Card p G q ˘ Card p F q “ p ` ε q Card p F q . Set c “ d { cd { P B ` l p G F q b M L b C p X q ˘ . Then, at the third step using (3.7) on the first term and (3.1) on the secondterm, ›› p c q ˚ d { bd { c ´ d { ad { ›› ď ›› d { c ˚ dbdcd { ´ d { c ˚ bdcd { ›› ` ›› d { c ˚ bdcd { ´ d { c ˚ bcd { ›› ` ›› d { c ˚ bcd { ´ d { ad { ›› ď } c }} dc ´ cd } ` } c ˚ bc ´ a } ă ˆ ˙ ` “ . Now, under our identification of a as an element in the crossed product, d { ad { is a diagonal operator on the Hilbert M L b C p X q -module l p G F qb M L b C p X q , and for any g P K , since ∆ p g q “
1, the g -th diagonal entry issimply p α g ´ b id M L qp a q . Likewise, as b is invariant under the group action, d { bd { is a diagonal matrix whose diagonal entries are scalar multiples ofthe constant projection b . We restrict all these diagonal entries (which arematrix valued functions on X ) to Y “ Ş g P F X K T ´ g p Y q .Let p P B ` l p G F q b M L b C p X q ˘ be the diagonal projection whosediagonal g -th entry is 1 if g P K and zero otherwise. We obtain ›› pd { ad { p ´ p p c q ˚ d { bd { c p ›› ă . This remains true after restricting to Y . Note that pd { ad { p “ pap .Thus, the projection pap | Y is Murray-von-Neumann subequivalent to the cutdown of b to B ` l p G F q b M L b C p X q ˘ , restricted to Y , which is aconstant projection of rank Card p G F q ¨ rank p b q . Notice that pap | Y is theprojection onto r E “ à g P F X K T ˚ g p E q| Ş h P F X K T ´ h p Y q . Now, c ` r E ˘ “ ! g P F X K p ` T ˚ g p η qq| Ş h P F X K T ´ h p Y q , so if r E ‘ E is a trivial bundle, then c p E q “ c ` r E ˘ ´ “ ! g P F X K p ´ T ˚ g p η qq| Ş h P F X K T ´ h p Y q . In particular, c k ¨ Card p F X K q{ p E q ‰
0, so rank p E q ě k ¨ Card p F X K q{ r E cannot embed into a trivial bundle of rank less than k ¨ Card p F X K q . Thus(3.9) Card p G F q ¨ rank p b q ě k ¨ Card p F X K q . Using (4) and (3.4) at the second step, k ¨ Card p F X K q Card p F q ě k ¨ Card p F q Card p F q ´ k ¨ Card p F r K q Card p F qą mcid k p T ; Q q ´ ε ´ kε , so, using (3.9) at the first step, (3.8) at the second step, and (3.3) at thethird step,rank p b q ě k ¨ Card p F X K q Card p F q ˆ Card p F q Card p G F q ˙ ą mcid k p T ; Q q ´ ε ´ kε ` ε ą mcid k p T ; Q q ´ ε . This is what we set out to prove.Now recall that m is the greatest integer with m ă mcid k p T ; Q q . Choose ε ą k p T ; Q q ´ m ´ ε ą
0. Make the choices above withthis value of ε , but take b to be a constant projection with rank p b q “ m .Then a  C p X q ⋊ T G b . For any invariant tracial state τ on C p X q , and hencefor any tracial state τ on C p X q ⋊ T G , we have d τ p b q “ rank p b q and d τ p a q ď rank p a q “ k {
2. So C p X q ⋊ T G does not have p m ´ k { q -comparison. Thus,rc p C p X q ⋊ T G q ě m ´ k { C p X q ⋊ T G is simple, then, since this algebra is also stably finite,Proposition 6.3 of [Tom06] implies that the set of real numbers r such that C p X q ⋊ T G has r -comparison is closed. So in fact rc p C p X q ⋊ T G q ą m ´ k { (cid:3) Applying this result to the subshift of [Dou17] and rounding several esti-mates, we get the following result. The “loss factor” 1 ´ ´ ρρ , which is themain part of the difference between our estimate and the conjectured valueof rc p C p X q ⋊ T G q , is 1 if ρ “ ρ is close to 1, but makesthe estimate useless if ρ ď . If we assume q H k p Z ; Q q ‰
0, we can removethis factor. See Corollary 4.6.
EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 19
Corollary 3.4.
Let G be a countable amenable group, let Z be a polyhedron,and let ρ P p , q . Let p X, T q be the minimal subshift of the shift on Z G constructed in Section 4 of [Dou17] to satisfy mdim p T q “ dim p Z q ρ . Then rc p C p X q ⋊ T G q ą
12 mdim p T q ˆ ´ ´ ρρ ˙ ´ . Proof.
We may certainly assume dim p Z q ą
0. Let k be largest even integerwith k ă dim p Z q , so that dim p Z q´ ď k ď dim p Z q´
1. Then mcid k p T q ě kρ by Proposition 2.6. Therefore rc p C p X q ⋊ T G q ą kρ ´ ´ k { k “ dim p Z q ´
2, using ρ ď p C p X q ⋊ T G q ą r dim p Z q ´ s ρ ´ ´ dim p Z q ´ “
12 dim p Z q ρ ˆ ´ ρ ˙ ´ ρ ě
12 dim p Z q ρ ˆ ´ ´ ρρ ˙ ´ . If k “ dim p Z q ´
1, then insteadrc p C p X q ⋊ T G q ą
12 dim p Z q ρ ˆ ´ ´ ρρ ˙ ´ ρ ´ ą
12 dim p Z q ρ ˆ ´ ´ ρρ ˙ ´ . Now substitute dim p Z q ρ “ mdim p T q . (cid:3) Example 3.5.
For every infinite countable amenable group G and every N ą
0, there is a cube Z and a minimal subsystem p X, T q of the shift actionof G on Z G such that rc p C p X q ⋊ T G q ą N .In Corollary 3.4 take ρ “ , and choose an integer d with d ą p N ` q .The subshift T in Section 4 of [Dou17] satisfies mdim p T q “ dρ , sorc p C p X q ⋊ T G q ą dρ ˆ ´ ´ ρρ ˙ ´ “ d ´ ą N. Symmetric mean cohomological independence dimension
In this section, we define a variant of mean cohomological independencedimension, which we use to obtain sharper lower bounds on the radius ofcomparison for certain subshifts.We recall the elementary symmetric polynomials: for n, r
P t , , , . . . u with r ď n , σ r p x , x , . . . , x n q “ ÿ ď j ă j 㨨¨ă j r ď n x j x j ¨ ¨ ¨ x j r is the elementary symmetric polynomial of degree r in the n variables x , x , . . . , x n . Definition 4.1.
Let k be an even integer, let Y be a compact metrizablespace, let R be a commutative unital ring, let F be a finite set, and let r P t , , , . . . , Card p F qu . If p η g q g P F is a family of elements in q H k p Y ; R q indexed by F , we define σ r ` p η g q g P F ˘ as follows. Set n “ Card p F q , andenumerate F as t g , g , . . . , g n u . Then define σ r ` p η g q g P F ˘ “ ÿ ď j ă j 㨨¨ă j r ď n η g j ! η g j ! ¨ ¨ ¨ ! η g jr . We call it the r -th elementary symmetric polynomial of p η g q g P F .Since k is even and R is commutative, σ r ` p η g q g P F ˘ does not depend onthe enumeration of F . Definition 4.2.
Let X be a compact metrizable space, let G be a countableamenable group, and let T be an action of G on X . Let R be a commutativeunital ring. For any even integer k , we take smcid k p T ; R q to be the largest d P r , such that the following happens.There are a finite open cover U of X such that D X p U q P t k, k ` u and η P q H k p X ; U ; R q (Definition 1.8: ˇCech classes using U ) such that for everyfinite subset G Ă G and every ε ą p G , ε q -invariant nonemptyfinite set F Ă G and r P t , , , . . . , Card p F qu for which:(1) Following Definition 4.1, we have σ r ` p T ˚ g p η qq g P F ˘ ‰ kr Card p F q ą d ´ ε .We then say that T has symmetric mean k -th cohomological independencedimension d with coefficients in R .We define the symmetric mean cohomological independence dimension smcid p T ; R q to be the supremum of smcid k p T ; R q over all even k P N .As we will see, this definition is sometimes useful for subshifts of theshift on Z G when q H k p Z ; Q q ‰
0. It doesn’t give anything useful for theshift on pr , s d q G . One would like to ask that for every ε ą Y Ă X , a finite open cover U of Y in X such that D Y p U q Pt k, k ` u , and η P q H k p Y ; U ; R q such that for every finite subset G Ă G and every δ ą p G , δ q -invariant nonempty finite set F Ă G and r P t , , , . . . , Card p F qu for which (1) and (2) hold. This notion canbe used on the full shift, but does not seem useful for minimal systems. Theset Ş g P G T ´ g p Y q is closed and G -invariant. If it is X , then Y “ X . If itis ∅ , then there is a finite subset F Ă G such that Ş g P F T ´ g p Y q “ ∅ , whichspoils (2).We could give a generalization of this definition, possibly useful for non-minimal systems, by considering the supremum of the values of smcid k p T Y ; R q as Y ranges over all closed invariant subsets of X ; we chose to avoid it herein order to lighten notation. Lemma 4.3.
In the situation of Definition 4.2, we have smcid k p T ; R q ď mcid k p T ; R q .Proof. The quantity in Definition 1.11 cannot become larger when we imposethe restriction Y “ X . Given that restriction, the inequality to be provedfollows from the fact that if σ r ` p T ˚ g p η qq g P F ˘ ‰
0, then at least one of itsterms must be nonzero, that is, there is a subset F Ă F with Card p F q “ r and such that the cup product of T ˚ g p η q over all g P F is nonzero. (cid:3) In the following proposition, the assumption that R is a principal idealdomain is needed in order to use the K¨unneth Formula. Proposition 4.4.
Let G be a countable amenable group. Let R be a principalideal domain. Let k be an even integer, and let Z be a finite CW-complex EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 21 with dim p Z q P t k, k ` u and q H k p Z ; R q ‰ . Let X be a closed G -invariantsubset of Z G , and let T be the restriction to X of the shift action of G on Z G .Suppose that there is an X -unconstrained subset of G (Definition 2.4) withdensity at least ρ (Definition 2.3). Then smcid k p T ; R q ě kρ .Proof. Let J Ă G be X -unconstrained with witness z and δ p J q ě ρ . Bytranslation, we may assume without loss of generality that 1 P J . If kρ “ kρ ą
0. In particular, k ě V of Z such that q H k p Z ; V ; R q ‰
0, and let η P q H k p Z ; V ; R q be nonzero. Let q : Z G Ñ Z be the projection onto the coordi-nate g “
1. Then q p X q “ Z . Set U “ q ´ p V q X X , and set η “ p q | X q ˚ p η q .We claim that for any finite set F Ă G , if we set r “ Card p F X J q , then σ r ` p T ˚ g p η qq g P F ˘ ‰
0. To prove the claim, first set n “ Card p F q . Definemaps t : Z F X J Ñ X and p : X Ñ Z F as follows. Take p to be the restriction of the projection map Z F ˆ Z G r F Ñ Z F . For x P Z F X J define t p x q g “ x g g P F X Jz g g P G r p F X J q . The fact that z is a witness for the unconstrainedness of J implies that t p x q as defined here really is in X .Enumerate the elements of F as g , g , . . . , g n , with g , g , . . . , g r P F X J .The K¨unneth Formula for ˇCech cohomology (applied to finite CW com-plexes) gives an injective unital ring homomorphism ι : n â j “ q H ˚ p Z ; R q Ñ q H ˚ p Z F ; R q such that, with q j : Z F Ñ Z being the projection to the g j -th coordinate,and with µ j P q H ˚ p Z ; R q for j “ , , . . . , n , we have ι p µ b µ b ¨ ¨ ¨ b µ n q “ q ˚ p µ q ! q ˚ p µ q ! ¨ ¨ ¨ ! q ˚ n p µ n q . Similarly, we get an injective unital ring homomorphism ι : r â j “ q H ˚ p Z ; R q Ñ q H ˚ p Z F X J ; R q . For j “ , , . . . , n , set λ j “ q ˚ j p η q “ ι p b b ¨ ¨ ¨ b b η b b ¨ ¨ ¨ b q , with η in position j . We have σ r p λ , λ , . . . , λ n q “ ÿ ď j ă j 㨨¨ă j r ď n λ j ! λ j ! ¨ ¨ ¨ ! λ j r . Let j P t r ` , r ` , . . . , n u (the set indices corresponding to the elementsof F r p F X J q ). Then, since q j ˝ p ˝ t is the constant map with value z g j ,we have p p ˝ t q ˚ p λ j q “ p q j ˝ p ˝ t q ˚ p η q “ . Therefore p p ˝ t q ˚ ` σ r p λ , λ , . . . , λ n q ˘ “ p p ˝ t q ˚ p λ ! λ ! ¨ ¨ ¨ ! λ r q“ ι p η b η b ¨ ¨ ¨ b η q (4.1)(with r tensor factors in the last expression). Since η b η b ¨ ¨ ¨ b η ‰ σ r ` p T ˚ g p η qq g P F ˘ “ p ˚ ` σ r p λ , λ , . . . , λ n q ˘ . So t ˚ ` σ r ` p T ˚ g p η qq g P F ˘˘ ‰
0, whence σ r ` p T ˚ g p η qq g P F ˘ ‰
0, as claimed.We have D Z p V q ď k ` p Z q ď k `
1, and it follows that D Z p U q ď k `
1. Since kρ ą
0, we have J ‰ ∅ . Choosing any finitesubset F Ă G with F X J ‰ ∅ , the claim certainly implies η ‰
0. Since η P q H k p X ; U ; R q , this implies D Z p U q ě k .Now let G Ă G be finite and let ε ą
0. It follows from Definition 2.3that there is a nonempty finite p G , ε q -invariant subset F Ă G such thatCard p J X F q Card p F q ą ρ ´ εk . Then, using the claim for the second equation, the number r “ Card p J X F q satisfies kr Card p F q ą kρ ´ ε and σ r ` p T ˚ g p η qq g P F ˘ ‰ . This completes the proof. (cid:3)
Theorem 4.5.
Let G be a countable amenable group, let X be a compactmetrizable space, and let T be an action of G on X . Let k be an even integer.Let m be the greatest integer with m ă smcid k p T ; Q q . Then rc p C p X q ⋊ T G q ě m . If C p X q ⋊ T G is simple, then rc p C p X q ⋊ T G q ą m . In particular, rc p C p X q ⋊ T G q ě smcid k p T ; Q q ´ Proof of Theorem 4.5.
Fix an even integer k . Fix ε ą U of X such that D X p U q P t k, k ` u and acohomology class η P q H k p X ; U ; Q q such that for any δ ą G Ă G there exist a nonempty finite p G , δ q -invariant subset F and r P t , , , . . . , Card p F qu satisfying:(1) Following Definition 4.1, we have σ r ` p T ˚ g p η qq g P F ˘ ‰ kr Card p F q ą smcid k p T ; Q q ´ δ .Arguing as in the proof of Theorem 3.3, after possibly replacing η by anonzero scalar multiple of itself, we can choose a vector bundle E over X such that rank p E q “ k { c p E q “ ` η .Let L be the dimension of some trivial bundle which has E as a direct sum-mand, and let a P M L p C p X qq be the projection onto E . Let b P M p C p X qq be a constant projection. Suppose that a - C p X q ⋊ T G b . We are going toprove that this implies rank p b q ě p smcid k p T ; Q q ` k q ´ ε .Arguing again as in the proof of Theorem 3.3 (in particular, possiblyincreasing L ), we may assume that b P M L p C p X qq and that there is c P EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 23 M L p C p X q ⋊ T G q in the algebraic crossed product such that(4.2) } c ˚ bc ´ a } ă . With u g P C p X q ⋊ T G being the standard unitary corresponding to g P G ,there are a finite subset G Ă G and elements c g P M L p C p X qq for g P G such that c “ ř g P G c g p M L b u g q . Choose δ ą δ ¨ p} c } ` q ¨ ÿ g P G } c g } ă . We proceed to construct a cutoff function, where here we need to makechoices which are a bit different than those used in the proof of Theorem3.3. Choose a nonempty finite p G , δ q -invariant subset G of G with 1 P G .Choose ε ą k p T ; Q q ` k ´ ε ` ε ą smcid k p T ; Q q ` k ´ ε . Set ε “ min ˆ ε , ε p G q ˙ . Find a finite nonempty ` G Y G , ε ˘ -invariant subset F Ă G and r Pt , , , . . . , Card p F qu satisfying(4.4) σ r ` p T ˚ g p η qq g P F ˘ ‰ kr Card p F q ą smcid k p T ; Q q ´ ε . Set S “ ď h P G h ´ F. Then Card p S r F q ď ÿ h P G Card p h ´ F r F q “ ÿ h P G Card p F r hF qă Card p G q ε Card p F q ď ε Card p F q . (4.5)Let ∆ : G Ñ r , s be the function∆ “ p G q p χ G ˚ χ S q . (We have replaced F in the proof of Theorem 3.3 with S .) By the samereasoning as there, for t P G and g P G we have(4.6) | ∆ p t ´ g q ´ ∆ p g q| ď Card ` p tG r G q Y p G r tG q ˘ Card p G q ă δ. Likewise, by similar reasoning to that in the proof of Theorem 3.3, we seethat that ∆ p g q “ g P F .As in the proof of Theorem 3.3, let α be the corresponding action of G on C p X q , and view C p X q ⋊ T G as embedded in B p l p G q b C p X qq in the sameway as there. We define a multiplication operator d P B p l p G q b C p X qq by p d ξ qp g q “ ∆ p g q ξ p g q for g P G . By (4.6), for any t P G , } u t d u ˚ t ´ d } ă δ , whence } u t d ´ d u t } ă δ . Set d “ d b M L . It follows that(4.7) } cd ´ dc } ă δ ÿ g P G } c g } ă p} c } ` q . Notice that supp p ∆ q “ G S . Thus, we can view dM L p C p X q ⋊ T G q d asincluded in B ` l p G S q b M L b C p X q ˘ , with l p G S q b M L b C p X q regardedas a Hilbert M L b C p X q -module. Since G S is a finite set, this is a matrixalgebra over C p X q .Since, in particular, F is p G , ε q -invariant, and since G S r F “ ď g,h P G p gh ´ F r F q Ă ď g,h P G “ g p h ´ F r F q Y p gF r F q ‰ , we have(4.8) Card p G S r F q ď p G q ε Card p F q ă ε Card p F q . Set c “ d { cd { P B ` l p G F q b M L b C p X q ˘ . Then, arguing as in the proof of Theorem 3.3, we have ›› p c q ˚ d { bd { c ´ d { ad { ›› ă . Under our identification of a as an element in the crossed product, d { ad { is a diagonal operator on the Hilbert M L b C p X q -module l p G F q b M L b C p X q , and for any g P F , since ∆ p g q “
1, the g -th diagonal entry is sim-ply p α g ´ b id M L qp a q . Likewise, as b is invariant under the group action, d { bd { is a diagonal matrix whose diagonal entries are scalar multiples ofthe constant projection b .Let p P B ` l p G S q b M L b C p X q ˘ be the diagonal projection whose di-agonal g -th entry is 1 if g P F and zero otherwise. We obtain ›› pd { ad { p ´ p p c q ˚ d { bd { c p ›› ă . Note that pd { ad { p “ pap . Thus, the projection pap is Murray-von-Neumann subequivalent to the cutdown of b to B ` l p G S q b M L b C p X q ˘ ,which is a constant projection of rank Card p G S q ¨ rank p b q . Notice that pap is the projection onto r E “ À g P F T ˚ g p E q . Now, c ` r E ˘ “ ! g P F p ` T ˚ g p η qq , so if r E ‘ E is a trivial bundle, then c p E q “ c ` r E ˘ ´ “ ! g P F p ´ T ˚ g p η qq “ Card p F q ÿ j “ p´ q j σ j ` p T ˚ g p η qq g P F ˘ . In particular, c kr { p E q “ p´ q r σ r ` p T ˚ g p η qq g P F ˘ ‰ , so rank p E q ě kr {
2. Therefore r E does not embed in a trivial bundle of rankless than kr ` rank ` r E ˘ “ k p r ` Card p F qq . EAN COHOMOLOGICAL INDEPENDENCE DIMENSION 25
So Card p G S q ¨ rank p b q ě k p r ` Card p F qq , whence, using (4.8) at the secondstep, (4.4) at the fourth step, and (4.3) at the fifth step,rank p b q ě k ˆ r ` Card p F q Card p G S q ˙ ą k ˆ r ` Card p F qp ` ε q Card p F q ˙ “ ˜ kr Card p F q ` k ` ε ¸ ą ˆ smcid k p T ; Q q ´ ε ` k ` ε ˙ ą ` smcid k p T ; Q q ` k ˘ ´ ε , as wanted.Now recall that m is the greatest integer with m ă smcid k p T ; Q q . Choose ε ą smcid k p T ; Q q ´ ε ą m . In the argument above, usethis value of ε , and take b to be a constant projection with rank p b q “ m ` k . Then a  C p X q ⋊ T G b . For any invariant tracial state τ on C p X q , andhence for any tracial state τ on C p X q ⋊ T G , we have d τ p b q “ rank p b q and d τ p a q “ rank p a q “ k {
2. So C p X q ⋊ T G does not have m -comparison. Thus,rc p C p X q ⋊ T G q ě m .The argument for rc p C p X q ⋊ T G q ą m when C p X q ⋊ T G is simple is thesame as in the proof of Theorem 3.3. (cid:3) Corollary 4.6.
Let k be a strictly positive even integer, let Z be a k -dimensional polyhedron such that q H k p Z ; Q q ‰ , let G be a countable amen-able group, and let ρ P p , q . Let p X, T q be the minimal subshift of the shifton Z G constructed in Section 4 of [Dou17] to satisfy mdim p T q “ kρ . Then rc p C p X q ⋊ T G q ą mdim p T q ´ .Proof. Let J Ă G be as in Proposition 2.5. This proposition implies thatthe hypotheses of Proposition 4.4 are satisfied, so smcid k p T ; Q q ě kρ . The-orem 4.5 now gives rc p C p X q ⋊ T G q ą kρ ´
1. As in [Dou17], we havemdim p T q “ kρ . (cid:3) In particular, if k is even, then the subshifts of p G, p S k q G q in [Dou17]satisfy rc p C p X q ⋊ T G q ą mdim p T q ´
1. This is within 1 of the conjecturedvalue of rc p C p X q ⋊ T G q .5. Concluding remarks
For minimal subshifts as in Section 4, we can bound the radius of com-parison of the crossed product rc p C p X q ⋊ T G q from below by the largestinteger smaller than kρ {
2, where ρ is the density as in that example. Forsuitable choices of ρ , this can be arbitrarily close to kρ {
2. The fact that weonly get integers is a consequence of our method of proof. We know of noreason not to believe that the following may have an affirmative answer.
Question 5.1.
Let T be a topologically free and minimal action of a count-able amenable group G on a compact metrizable space X . Do we have12 mdim p T q ě rc p C p X q ⋊ T G q ě
12 mcid p T ; Q q ?For the lower bound, minimality does not seem to be relevant. In Proposition 2.6, we proved that if p X, G q is a subshift of Z G which hasan X -unconstrained set J Ă G with density ρ , thenmcid p T | X ; R q ě ρ Z dim p Z q ´ ^ . Proposition 2.8 of [Kri09] gives a related estimate with mean dimension inplace of mean cohomological independence dimension. While this does notshow that they coincide, it does mean that for reasonable spaces they are notfar apart, and suggests that they may coincide under reasonable conditions.
Question 5.2.
Does mean cohomological independence dimension coincidewith mean dimension for subshifts under the hypotheses of Proposition 2.6?Does this depend on the ring of coefficients?The space Z in Proposition 2.6 is a finite CW-complex. Bad spaces maywell behave quite differently. Scattered in [Dra05], one can find variousexamples of strange behavior of cohomological dimension in products, dif-ferent for different coefficient rings, and differences between cohomologicaldimension and covering dimension. We don’t know the mean cohomologicalindependence dimension of shifts or subshifts on badly behaved spaces. (Forthe mean dimension of shifts on finite dimensional badly behaved spaces,see [Tsu19]. That paper leaves open the mean dimension of the shift on, forexample, a compact space X with dim p X q “ 8 but integer cohomologicaldimension dim Z p X q “
3. Shifts on such spaces are perhaps more likely toexhibit strange behavior of mean cohomological independence dimension.)
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Department of Mathematics, Ben Gurion University of the Negev,P.O.B. 653, Be’er Sheva 84105, Israel
E-mail address : [email protected]@math.bgu.ac.il